Math Cartoons for Inverse Functions

The evolution of my lesson planning over the past three years goes something like this (By the way, instead of using evolution at first I used evolvement, which is not a word. Spell check suggested the word enslavement... ):

Year 1: The challenge was the unmerciful pace of time as I searched each night for the perfect way to teach a concept I was to teach the next day. Being frustrated  not finding it, I would organize a lesson or create some sort of activity to teach the concept, finish the lesson just in  time, execute the lesson in class, and would think either "Wow- not bad" or "Wow- that did not go at all like I envisioned".

Year 2: The challenge was looking over the lessons and asking "How can I better explain this?" and trying to improve my explanations to make the objectives of the lessons more achievable for my students.

Year 3: The challenge is realizing the way I have taught needs a major make-over. My goals to be the perfect explainer and presenter of information have been flawed. Understanding by Design has been the focus of our professional development, and this year I struggle to change the way I teach so students have joy in discovering and truly understanding concepts. (I just read a great blog at http://bowmandickson.com/2012/09/29/let-them-figure-out-the-power-rule/ concerning this.)

With that rant out of the way, I would like to say I have seen success with some of my units throughout the years. 

I am teaching logarithms at the moment. It has become one of my favorite units to teach. I start the unit with compositions and inverse functions. I find my students really have always enjoyed this lesson. The cause of the enjoyment- a cartoon. I know, I know a happy go-lucky cartoon is not the answer of true understanding, but it helps. I thought I would share it:

The one big observation I get EVERY YEAR from this cartoon is some students points out or shouts out "Where is the robots head?" referring to the third image along slide one. Again the thought through my head is "Wow. That did not go at all like I envisioned." 

However, this really helps with the understanding of why the inverse functions have x and y values and roles that swap. Everything our original machine/function does onto an input is undone by the inverse function. The output of the original function is the inverse's input because the inverse swoops in and takes it. The output of the inverse is what the original had started out with and worked to hard on to change.

Whatever it takes to make it stick, right? 

I have attached the PowerPoint I have for this lesson in the "Lesson's on the Loose" page of my blog. I also will be adding my exponential and logarithm lessons because this lesson on inverses launches that unit.